As an extension of the authors' work on isotropic vortical turbulence interacting with a shock wave (Lee, Lele & Moin 1993), direct numerical simulation and linear analysis are performed for stronger shock waves to investigate the effects of the upstream shock-normal Mach number (M1). A shock-capturing scheme is developed to accurately simulate the unsteady interaction of turbulence with shock waves. Turbulence kinetic energy is amplified across the shock wave, and this amplification tends to saturate beyond M1 = 3.0. An existing controversy between experiments and theoretical predictions on length scale change is thoroughly investigated through the shock-capturing simulation: most turbulence length scales decrease across the shock, while the dissipation length scale (ρq3/ε) increases slightly for shock waves with M1<1.65. Fluctuations in thermodynamic variables behind the shock wave are nearly isentropic for M1<1.2, and deviate significantly from isentropy for the stronger shock waves, due to the entropy fluctuation generated through the interaction.